Strength of Materials
Strength of Materials
This domain covers stress–strain, beams, torsion, columns, and combined loading. Free-body diagrams and boundary conditions decide most answers.
Core rules
- Axial: σ = Eε. Fully restrained thermal stress: σ = −EαΔT; the negative sign indicates compression on heating.
- Bending: σ = My/I; shear stress τ = VQ/(Ib). Deflection follows EI y″ = M. For a simply supported beam with central load P, maximum deflection is PL³/(48EI).
- Torsion: τ = Tr/J; angle of twist θ = TL/(GJ). For a solid circular shaft, J = πd⁴/32; for a hollow shaft, J = π(d_o⁴ − d_i⁴)/32.
- Columns: Euler buckling P_cr = π²EI/(L_e²); valid only when the buckling stress is below the proportional limit. Rankine–Gordon is used for intermediate slenderness.
Worked example
A solid shaft of diameter 50 mm carries 500 N·m. J = π(0.05)⁴/32 = 6.136 × 10⁻⁷ m⁴. Maximum shear stress τ = Tr/J = 500 × 0.025 / 6.136 × 10⁻⁷ = 20.4 × 10⁶ Pa = 20.4 MPa. The maximum shear occurs at the outer surface, r = d/2, not at the centre.
Traps
Using diameter in place of radius in Tr/J; confusing I and J; applying Euler's formula to a short column; ignoring the effective-length factor for different end conditions; and forgetting that thermal stress vanishes if expansion is free.